On the application of the Chebyshev-Bernstein method to a class of extremal functions satisfying some relationships, linear with respect to the coefficients

The aim of the article is to establish the form of nonnegative polynomials $y(x)$ of a power which does not exceed the given one, limiting the integral
$$\int_{-1}^1p(x)y_n(x)dx$$
where $p(x)$ is the given summed function, if the coefficients of the polynomial $y(x)$ satisfy any number s of linear relationships.